面白い is both that laugh-out-loud "funny" that you are looking for, and "hmm, interesting..." It functions as both meanings without losing the intensity of the first, like you were suggesting. It is totally about intonation, just like the word "funny" is in English.
おかしい is similar to the later meaning of "funny," as in strange or interesting, but it kind of suggests that something is wrong. For example, コンピュータがおかしくなった would be something like "my computer's acting funny/strange."
変 is a more extreme "weird." Like "abnormal."
So, the story that your friend tells you is 面白い, the way your computer is acting up is おかしい, and your little brother who likes to cosplay is 変.